We employ open source plane-wave code Quantum ESPRESSO [1]} for structural relaxation and phonon calculations of carbon defects in monolayer hBN. We use the optimized norm-conserving Vanderbilt (ONCV) pseudopotentials [2]} and a 55 Ry wavefunction energy cutoff. We choose a supercell size of \(6\times 6\)  which shows good convergence as tested in Ref. wu2019carrier, Tyler2021-err, Smart2021wk.
Charged defect correction is included to eliminate the spurious electrostatic interaction by using the techniques developed in Refs. PingJCP, wu2017first and implemented in the JDFTx code [3]}. Total energy, defect formation energy and geometry are obtained with the Perdew-Burke-Ernzerhof (PBE) exchange correlation functional [4]}. The charged defect formation energy \(E_f^q(d)\)  with the charge state \(q\)  is calculated by
\(E_f^q(d) = E_{\mathrm {tot}}^q(d) - E_{\mathrm {tot}}(p) - \sum _iN_i\mu _i + qE_{\mathrm {Fermi}} + E_{\mathrm {corr}},\) 
